Showing papers on "Bessel function published in 2001"

Analytic description of critical point nuclei in a spherical-axially deformed shape phase transition.

Abstract: An approximate solution at the critical point of the spherical to axially deformed shape phase transition in nuclei is presented. The eigenvalues of the Hamiltonian are expressed in terms of zeros of Bessel functions of irrational order.

Handbook of Mathematical Techniques for Wave/Structure Interactions

Abstract: INTRODUCTION The Water-Wave Problem The Linearised Equations Interaction of a Wave with a Structure Reciprocity Relations Energy of the Fluid Motion EIGENFUNCTION EXPANSIONS Introduction Construction of Vertical Eigenfunction Two-Dimensional Problems Three-Dimensional Problems Matched Eigenfunction Expansions MULTIPOLE EXPANSIONS Introduction Isolated Obstacles Multiple Bodies INTEGRAL EQUATIONS Source Distribution Green's Theorem Thin Obstacles Interior Problems Free-Surface Problems Numerical Evaluation of Green's functions Diffraction by a Gap in a Breakwater Diffraction by an Insular Breakwater Embedding Formulae Numerical Solutions THE WIENER-HOPF AND RELATED TECHNIQUES The Weiner-Hopf Technique Residue Calculus Theory ARRAYS The Wide-Spacing Approximation SMALL OBJECTS Introduction Breakwater with a Gap Vertical Cylinder Heaving Cylinder Eigenvalue Problems VARIATIONAL METHODS Scattering and Radiation Problems Eigenvalue Problems APPENDICES Bessel Functions Multipoles Principle Value and Finite Part Integrals

Knotted and linked phase singularities in monochromatic waves

Abstract: Exact solutions of the Helmholtz equation are constructed, possessing wavefront dislocation lines (phase singularities) in the form of knots or links where the wave function vanishes (‘knotted nothings’). The construction proceeds by making a nongeneric structure with a strength n dislocation loop threaded by a strength m dislocation line, and then perturbing this. In the resulting unfolded (stable) structure, the dislocation loop becomes an ( m , n ) torus knot if m and n are coprime, and N linked rings or knots if m and n have a common factor N ; the loop or rings are threaded by an m -stranded helix. In our explicit implementation, the wave is a superposition of Bessel beams, accessible to experiment. Paraxially, the construction fails.

Eigenvalues of the Laguerre Process as Non-Colliding Squared Bessel Processes

Abstract: Let $A(t)$ be an $n\times p$ matrix with independent standard complex Brownian entries and set $M(t)=A(t)^*A(t)$ This is a process version of the Laguerre ensemble and as such we shall refer to it as the Laguerre process The purpose of this note is to remark that, assuming $n > p$, the eigenvalues of $M(t)$ evolve like $p$ independent squared Bessel processes of dimension $2(n-p+1)$, conditioned (in the sense of Doob) never to collide More precisely, the function $h(x)=\prod_{i 0}$ is the Wishart process considered by Bru (1991) There it is shown that the eigenvalues of $M(t)$ evolve according to a certain diffusion process, the generator of which is given explicitly An interpretation in terms of non-colliding processes does not seem to be possible in this case We also identify a class of processes (including Brownian motion, squared Bessel processes and generalised Ornstein-Uhlenbeck processes) which are all amenable to the same $h$-transform, and compute the corresponding transition densities and upper tail asymptotics for the first collision time

Optical dipole traps and atomic waveguides based on Bessel light beams

Abstract: We theoretically investigate the use of Bessel light beams generated using axicons for creating optical dipole traps for cold atoms and atomic waveguiding. Zeroth-order Bessel beams can be used to produce highly elongated dipole traps, allowing for the study of one-dimensional trapped gases and realization of a Tonks gas of impenetrable bosons. First-order Bessel beams are shown to be able to produce tight confined atomic waveguides over centimeter distances.

A three-dimensional multifrequency large signal model for helix traveling wave tubes

Abstract: A three-dimensional (3D) multifrequency large signal model of the beam wave interaction in a helix TWT is described The beam is divided into a set of discrete rays, or "beamlets", instead of the disks or rings used in one-dimensional (1-D) or two-dimensional (2-D) models The RF fields supported by the helix are represented by a tape helix model that uses a modal expansion including the full (Bessel function) radial dependence of the fields; both forward and backward synchronous space harmonics are included in the model RF space charge fields are obtained from solutions of the Helmholtz equations for the RF electric and RF magnetic fields, using the beam current and charge densities as sources The dc space charge electric field is similarly obtained from a solution of Poisson's equation This model has been implemented in a code called CHRISTINE 3D, a generalization of the one dimensional CHRISTINE code The full three dimensional treatment permits the accurate computation of large signal gain and efficiency, taking into account the self-consistent variation of beam radius along the interaction space The code also computes helix interception current and transverse beam distributions at the entrance to the collector-important design data that are unavailable from a 1D model Results from the CHRISTINE 3D code are shown to compare very favorably with measurements of output power, efficiency, and interception current vs drive power Its predictions for spent beam distributions also compare very well with measurements Run times for the code are problem dependent, but for a single case of interest are typically 1 to 5 min on a 450 MHz PC, orders of magnitude shorter than that required for a comparable 3D particle-in-cell simulation

Generalized Harmonic Analysis and Wavelet Packets: An Elementary Treatment of Theory and Applications

Abstract: The book presents a more comprehensive treatment of transmutation operators associated with the Bessel operator, and explores many of their properties They are fundamental in the complete study of the Bessel harmonic analysis and the Bessel wavelet packets Many applications of these theories and their generalizations have been injected throughout the text by way of a rich collection of problems and references The results and methods in this book should be of interest to graduate and researchers working in special functions such as Fourier analysis, hypergroup and operator theories, differential equations, probability theory and mathematical physics Background materials are given in adequate detail to enable a graduate student to proceed rapidly from the very basics of the frontier of research in the area of generalized harmonic analysis and wavelets

Comments on "Exact solutions of electromagnetic fields in both near and far zones radiated by thin circular-loop antennas: a general representation" [with reply]

Abstract: This paper presents an alternative vector analysis of the electromagnetic (EM) fields radiated from thin circular-loop antennas of arbitrary radius a. This method, which employs the dyadic Green's function in the derivation of the EM radiated fields, makes the analysis more general, compact, and straightforward than those two methods published recently by Werner (1996) and Overfelt (1996). Both near and far zones are considered so that the EM radiated fields are expressed in terms of the vector-wave eigenfunctions. Not only the exact solution of the EM fields in the near and far zones outside the region (where r>a) is derived by the use of the spherical Hankel function of the first kind, but also the closed-series form of the EM fields radiated in the near zone inside the region 0/spl les/r

Axicon-based Bessel resonator: analytical description and experiment.

Abstract: We present a new scheme for an optical resonator for production of Bessel and Bessel–Gauss light beams. The resonator with Bessel modes is composed of two plane mirrors with an axicon placed close to one of them. If this mirror is concave, the modes are Bessel–Gauss light beams. Analytical expressions relating parameters of the resonator and characteristics of its modes are obtained and analyzed. The results are verified with the Fox–Li algorithm. The resonator scheme was implemented in an experiment to confirm the possibility of the generation of zero-order Bessel beams. It was found that multipass modes can also oscillate in the resonator if its apertures are large enough.

Finite two-dimensional oscillator: II. The radial model

Abstract: A finite two-dimensional radial oscillator of (N + 1)2 points is proposed, with the dynamical Lie algebra so(4) = su(2)x⊕su(2)y examined in part I of this work, but reduced by a subalgebra chain so(4)⊃so(3)⊃so(2). As before, there are a finite number of energies and angular momenta; the Casimir spectrum of the new chain provides the integer radii 0≤ρ≤N, and the 2ρ + 1 discrete angles on each circle ρ are obtained from the finite Fourier transform of angular momenta. The wavefunctions of the finite radial oscillator are so(3) Clebsch-Gordan coefficients. We define here the Hankel-Hahn transforms (with dual Hahn polynomials) as finite-N unitary approximations to Hankel integral transforms (with Bessel functions), obtained in the contraction limit N→∞.

Millimetre-wave Bessel beams using computer holograms

Abstract: A computer-generated binary amplitude hologram is used to transform an initial Gaussian electromagnetic field with spherical phase front at 310 GHz into a non-diffracting Bessel beam. The beam profile is measured with the help of a near field scanner. In contrast to the situation in the optical region, both amplitude and phase information is readily obtainable from the generated field.

New localized Superluminal solutions to the wave equations with finite total energies and arbitrary frequencies

Abstract: By a generalized bidirectional decomposition method, we obtain many new Superluminal localized solutions to the wave equation (for the electromagnetic case, in particular) which are suitable for arbitrary frequency bands; various of them being endowed with finite total energy. We construct, among the others, an infinite family of generalizations of the so-called "X-shaped" waves. [PACS nos.: 03.50.De; 41.20;Jb; 83.50.Vr; 62.30.+d; 43.60.+d; 91.30.Fn; 04.30.Nk; 42.25.Bs; 46.40.Cd; 52.35.Lv. Keywords: Wave equations; Wave propagation; Localized beams; Superluminal waves; Bidirectional decomposition; Bessel beams; X-shaped waves; Microwaves; Optics; Special relativity; Acoustics; Seismology; Mechanical waves; Elastic waves; Gravitational waves; Elementary particle physics].

Transformation of the order of Bessel beams in uniaxial crystals

Abstract: The optical transformation of a zeroth-order Bessel beam to a second-order Bessel beam is studied theoretically and experimentally in the case when the beam propa-gates along the optical axis of a uniaxial crystal. It is shown that, if the crystal length or the cone angle of the incident beam are chosen properly, the energy of the input field can be almost completely converted into a second-order Bessel beam.

Relationship between elegant Laguerre-Gauss and Bessel-Gauss beams.

Abstract: We show that the elegant Laguerre–Gauss light beams of high radial order n are asymptotically equal to Bessel–Gauss light beams. The Bessel–Gauss beam equivalent to each elegant Laguerre–Gauss beam is found and shown to have almost identical propagation factors M2. In the limit n→∞, elegant Laguerre–Gauss beams can be identified with Durnin’s Bessel beam. Our results suggest a new experimental procedure for generating light beams with nondiffractinglike properties directly from the output of a stable resonator.

The Askey-Wilson Function Transform Scheme

Abstract: In this paper we present an addition to Askey’s scheme of q- hypergeometric orthogonal polynomials involving classes of q-special functions which do not consist of polynomials only. The special functions are q-analogues of the Jacobi and Bessel function. The generalized orthogonality relations and the second order q-differenee equations for these families are given. Limit transitions between these families are discussed. The quantum group theoretic interpretations are discussed shortly.

Explicit zeta functions for bosonic and fermionic fields on a non-commutative toroidal spacetime†

Abstract: Explicit formulae for the zeta functions ζα(s) corresponding to bosonic (α = 2) and to fermionic (α = 3) quantum fields living on a non-commutative, partially toroidal spacetime are derived. Formulae for the most general case of the zeta function associated with a quadratic + linear + constant form (in Z) are obtained. They provide the analytical continuation of the zeta functions in relation to the whole complex s plane, in terms of series of Bessel functions (of fast, exponential convergence), thus being extended Chowla-Selberg formulae. As is well known, this is the most convenient expression that can be found for the analytical continuation of a zeta function; in particular, the residua of the poles and their finite parts are explicitly given. An important novelty is the fact that simple poles show up at s = 0, as well as in other places (simple or double, depending on the number of compactified, non-compactified and non-commutative dimensions of the spacetime) where they had never appeared before. This poses a challenge to the zeta-function regularization procedure.

New Improvements for Mie Scattering Calculations

Abstract: New improvements to compute Mie scattering quantities are presented. They are based on a detailed analysis of the various sources of error in Mie computations and on mathematical justifications. The algorithm developed on these improvements proves to be reliable and efficient, without size ($x=2\pi R/\lambda$) nor refractive index ($m=m_R-{\rm i}m_I$) limitations, and the user has a choice to fix in advance the desired precision in the results. It also includes a new and efficient method to initiate the downward recurrences of Bessel functions.

Eigenvalues of the laguerre process as non-colliding squared bessel processes

Abstract: Let A(t) be a np matrix with independent standard complex Brownian entries and set M(t )= A(t)A(t). This is a process version of the Laguerre ensemble and as such we shall refer to it as the Laguerre process. The purpose of this note is to remark that, assuming n p, the eigenvalues of M(t) evolve like p independent squared Bessel processes of dimension 2(n p +1 ), conditioned (in the sense of Doob) never to collide. More precisely, the function h(x )= Q i

The properties of sine, spherical Bessel and reduced Bessel functions for improving convergence of semi-infinite very oscillatory integrals: the evaluation of three-centre nuclear attraction integrals over B functions

Abstract: This paper focuses on the use of useful properties of sine, spherical Bessel and reduced Bessel functions to simplify the application of the nonlinear D- and -transformations for accelerating the convergence of semi-infinite very oscillatory integrals and to reduce the calculation times keeping a high predetermined accuracy. Three-centre nuclear attraction integrals, which are one of the most difficult type involved in density functional theory methods when using a basis set of B functions, are evaluated using the new approach. The numerical results show the efficiency of the new method compared with other alternatives.

Three-dimensional vibration analysis of a homogeneous transversely isotropic thermoelastic cylindrical panel

Abstract: In this paper, based on three-dimensional thermoelasticity, an exact analysis of the free vibrations of a simply supported, homogeneous, transversely isotropic, cylindrical panel is presented. Three displacement potential functions are introduced so that the equations of motion and heat conduction are uncoupled and simplified. It is noticed that a purely transverse mode is independent of temperature change and the rest of the motion. The equations for free vibration problems are further reduced to four second-order ordinary differential equations, after expanding the potential and temperature functions with an orthogonal series. A modified Bessel function solution with complex arguments is directly used for complex eigenvalues. To clarify the developed method and compare the results to the existing ones, numerical examples are presented.

Waves in wood: free vibrations of a wooden pole

Abstract: Elastic waves in materials with cylindrical orthotropy are considered, this being a plausible model for a wooden pole. For time-harmonic motions, the problem is reduced to some coupled ordinary differential equations. Previously, these have been solved using the method of Frobenius (power-series expansions). Here, Neumann series (expansions in Bessel functions of various orders) are used, motivated by the known classical solutions for homogeneous isotropic solids. This is shown to give an effective and natural method for wave propagation in cylindrically orthotropic materials. As an example, the frequencies of free vibration of a wooden pole are computed. The problem itself arose from a study of ultrasonic devices as used in the detection of rotten regions inside wooden telegraph (utility) poles and trees; some background to these applications is given.

Nonparaxial Bessel-Gauss beams.

Abstract: We study the nonparaxial propagation of Bessel–Gauss beams of any order. Closed-form expressions of all corrections to be added to the solution that is pertinent to the corresponding paraxial problem are found. Such corrections are expressed in terms of two families of polynomials, defined through recurrence rules, that encompass the Laguerre–Gauss polynomials for the particular case of a fundamental Gaussian beam. Numerical examples are shown.

Subspaces with normalized tight frame wavelets in R

Abstract: In this paper we investigate the subspaces of L 2 (R) which have normalized tight frame wavelets that are defined by set functions on some measurable subsets of R called Bessel sets. We show that a subspace admitting such a normalized tight frame wavelet falls into a class of subspaces called reducing subspaces. We also consider the subspaces of L 2 (R) that are generated by a Bessel set E in a special way. We present some results concerning the relation between a Bessel set E and the corresponding subspace of L 2 (R) which either has a normalized tight frame wavelet defined by the set function on E or is generated by E.

An extremely efficient approach for accurate and rapid evaluation of three-centre two-electron Coulomb and hybrid integrals over B functions

Abstract: Analytic expressions of three-centre two-electron Coulomb and hybrid integrals over B functions are obtained using the Fourier transform method thoroughly explored by Steinborn's group. These analytic expressions involve semi-infinite integrals which are slowly convergent due to the presence of hypergeometric and spherical Bessel functions in the integrands. We have proven that these hypergeometric functions can be expressed as finite expansions and the integrands involving these series satisfy all the conditions required to apply the H approach which greatly simplifies the application of the nonlinear -transformation. This work presents a rapid and accurate evaluation of these integrals, obtained by using a new approach, which we called S. This new method is based on the H and methods and some practical properties of spherical Bessel, reduced Bessel and sine functions. The S method has greatly simplified the calculations, avoiding the long and difficult implementation of the successive zeros of the spherical Bessel function and a method for solving linear systems, which are required by H and .